Uniform pro-p group

A finitely generated powerful p-group in the category of pro-p groups is uniform if all successive indices in its lower p-series are equal. Equivalently it is torsion-free. If its generator number is $d$, the successive indices are $p^d$, the power map carries $G^{p^n}$ bijectively onto $G^{p^{n+1}}$, and an ordered minimal generating set gives unique coordinates $g_1^{{\lambda }_1}\cdots g_d^{{\lambda }_d}$ with ${\lambda }_i$ in the p-adic integers. For $p=2$, powerfulness means $[G,G]\le G^4$.

For an odd prime $p$, the matrix logarithm and matrix exponential are inverse maps between $1+p{M}_d({\mathbb{Z}}_p)$ and $p{M}_d({\mathbb{Z}}_p)$. They imply that the $p$th-power map carries $U_m$ bijectively onto $U_{m+1}$. Direct multiplication gives $[U_i,U_j]\subseteq U_{i+j}$, so the group is a powerful pro-p group. The quotient $U_m/U_{m+1}$ is the additive matrix space over ${\mathbb{F}}_p$, with constant index $p^{d^2}$. The $d^2$ matrices $1+p{E}_{ij}$ topologically generate, by successive approximation through these quotients. The lower p-series is exactly $U_m$, so the group is a uniform pro-p group and a torsion-free group. At $p=2$ the first congruence subgroup contains $-I$ and is not uniform.

For odd $p$ and a finitely generated powerful pro-p group, put $U_j=G^{p^j}$. The displayed group homomorphisms are surjective. If one has a nonzero kernel, choose $x\in U_j\setminus U_{j+1}$ with $x^p\in U_{j+2}$. Successive corrections $x\mapsto x{u}_r$, with $u_r\in U_r$, cancel its $p$-th power one layer at a time, using $(x{u}_r{)}^p\equiv x^p{u}_r^p(\mathrm{mod}{U}_{r+2})$. Compactness and the separated power filtration give a nonidentity limit of order $p$. Thus a torsion-free group has constant layer indices and is a uniform pro-p group. Conversely constant indices make every displayed map injective; comparison at the first differing power layer proves that $x\mapsto x^p$ is injective, so there is no nontrivial torsion.

Every uniform pro-p group embeds continuously in ${\mathrm{GL}}_n({\mathbb{Z}}_p)$ for some finite $n$. Apply the Ado theorem to ${\mathbb{Q}}_p\otimes L_G$ and choose a deep power lattice whose representing matrices lie in $p\mathrm{End}(\Lambda )$, or $4\mathrm{End}(\Lambda )$ at $p=2$. The matrix exponential and logarithm give a faithful representation of the corresponding open normal power subgroup. Induce that representation to the whole group using its finitely many cosets. An element outside the open subgroup permutes the coset blocks nontrivially; an element inside it is detected by the original representation. Thus the induced representation is faithful. All block matrices preserve a finite free ${\mathbb{Z}}_p$-module, giving the integral embedding.

For a uniform pro-p group with $U_i=G^{p^i}$, the map $x\mapsto x^{p^a}$ transports congruences byCommutator collection gives $(yu{)}^p\equiv y^p{u}^p(\mathrm{mod}{U}_{b+2})$ for $u\in U_b$, and the power map is an isomorphism $U_b/U_{b+1}\to U_{b+1}/U_{b+2}$. Thus a nontrivial first difference moves down exactly one layer when taking a $p$th power. Iterating gives the equivalence; completeness supplies unique roots when the congruence is read in reverse.

For a uniform pro-p group, the power map $x\mapsto x^{p^n}$ is a homeomorphism onto $G^{p^n}$. Transport multiplication back to $G$ byUnique roots make this an associative group operation. For odd $p$, group commutator collection gives $x{+}_{n}y\equiv y{+}_{n}x(\mathrm{mod}{G}^{p^{n+1}})$ and $x{+}_{n}y\equiv x{+}_{n-1}y(\mathrm{mod}{G}^{p^n})$. The operations converge to addition in the Lie algebra of a uniform pro-p group.

The Lie algebra of a uniform pro-p group construction and BCH convergence on a powerful Lie lattice are mutually inverse on objects and morphisms. On the group side morphisms are continuous group homomorphisms; on the lattice side they are linear Lie algebra homomorphisms over the p-adic integers. A group homomorphism respects powers, roots and limits, and a Lie homomorphism respects every convergent Baker--Campbell--Hausdorff formula term.

The Lie algebra of a uniform pro-p group construction and BCH convergence on a powerful Lie lattice are mutually inverse on objects and morphisms. On the group side morphisms are continuous group homomorphisms; on the lattice side they are linear Lie algebra homomorphisms over the p-adic integers. A group homomorphism respects powers, roots and limits, and a Lie homomorphism respects every convergent Baker--Campbell--Hausdorff formula term.